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A287012 Number of time intervals that can be measured off with n ropes and a lighter.

%I A287012 #15 Aug 02 2017 13:09:37
%S A287012 2,6,15,34,78,174,386,844,1837,3960,8513,18238
%N A287012 Number of time intervals that can be measured off with n ropes and a lighter.
%C A287012 Suppose you have n ropes and a lighter. Each rope burns at a nonconstant rate but takes exactly one hour to burn completely from one end to the other. You can only light the ropes at either of their ends but can decide when to light each end as you see fit. If you're strategic in how you burn the ropes, how many specific lengths of time can you measure? (For example, if you had one rope, you could measure two lengths of time: one hour, by simply burning the entire rope from one end, and half an hour, by burning the rope from both ends and marking when the flames meet.)
%C A287012 In this sequence, the time intervals begin when any rope (or safety fuse, or match cord) begins or stops burning.
%H A287012 Mark Rickert, <a href="/A287012/a287012.py.txt">Python implementation</a>
%e A287012 a(2)=6: (i) Generate 1 by burning one rope from one end. (ii) Generate 2 by burning one rope from one end at t=0 and the other afterwards at t=1 from one end. (iii) Generate 1/2 by burning 1 rope from both ends. (iv) Generate 3/2 by burning 1 rope from one end at t=0 then the other from both ends at t=1 (or swapped order). (v) Generate 3/4 by burning one rope at t=0 from both ends, starting the other also at t=0 at one end, and lighting the other's second end at t=1/2 when the first rope's flames have met, so the 2nd rope's flames finish at t=3/4. (vi) Generate 1/4 using the technique for 3/4 and measuring the time between t=1/2 and t=3/4.
%e A287012 For n = 3 the a(3) = 15 solutions are 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, 1, 9/8, 5/4, 3/2, 7/4, 2, 5/2, 3.
%Y A287012 Cf. A283075, A188545.
%K A287012 nonn,hard,more
%O A287012 1,1
%A A287012 _Mark Rickert_, May 17 2017